Run StardewModdingAPI.exe in your main Stardew Valley folder. This will load the mods and then start the game.

As you progress in the Skull Cave, your level will be saved every X floors, where X is defined by SaveLevelEveryXFloors in the config. By default, this is 5, so your level will be saved once you pass floor 5, 10, 15, etc. It saves if you've made it to that level or past it, so you can fall down holes and not worry about skipping a save level. For example, falling from level 6 to level 11 without specifically hitting level 10 will still save your last level as 10.

To find your config file, look in the psconfigs folder within your mod folder (full path %appdata%\StardewValley\SkullCaveSaver\psconfigs. There you will find a file that looks like [YOUR SAVE NAME]_[LOTS OF NUMBERS].json Inside are the following options:

"LastMineLevel": 0,
"SaveLevelEveryXFloors": 5

LastMineLevel, which defaults to 0. This is the deepest level of the mine that the mod has saved. Edit this only if you want to skip ahead a number of levels. The skull cave starts on level 121.
SaveLevelEveryXFloors, which defaults to 5. This is how often the mod will save your cave level. It defaults to 5, like in the mine, where the elevator appears every five levels. You can set this as low as 1 to save your progress every level.